Computing-table.



C. H. WALLACE.

COMPUTING TABLE.

APPLICATION FILED MAR.15. 1911.

Patented Mzu. 25,-1919.

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COMPUTING TABLE.

111112111111011 FILED MAR. 15. 1911.

1,298,267. Patented Mar. 25,1919.

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C. H. WALLACE.

COMPUTING TABLE.

APPLICATION FILED MAR.15.1911.

1,298,267. Patented Mm. 1919.

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Powers of 2. Powers of 1/2.

c. H. WALLACE. COMPUTING TABLE.

APPLICATION FILED MAR. I5, I9I7.

Panfeamm. 2541/912937v L 'time To a-ZZ whom z't may concern:

""' VrBe it known that LOLABENc-E WAL- i'e'cn, a citizen of the United States, residing at St. Helena, in the county of- Napa and State of California, have invented certain y lnew and useful Improvements in Computrv"ing-Tables; and I do hereby declare the following to be a full, clear, and exact description of the invention, such as will en? able others Skilled in the art to which' it 1 Lappertains to make and use the same, refertvence being had to the accompanying draw- ?ing's, and to the letters and figures of refer- "ence marked thereon, this specification.

Thisinvention relates to new and useful *improvements in apparatus for use in multiplying and dividing numbers by addition,

forated wit holes for t e reception of pegs 2- and in' the provision of a longitudinally movable slide mounted in a slot inthe surface of the board and containin holes for the reception of pegs, the boar being diy vided by horizontal, transverse and diagonal lines .wlth peg receiving holes at the points pegs, the board being divided by horizontal,

of intersectlon.

the accompanying drawings and then speld-igs.A 5 and 6 are planviews showing ,tal yof figures. A `It is well known that an number may A"is eijual to the sum of 16 which is the fourth power of 2 plus 8 which is the third power.

. of 2 plus 1 which is the 0 power of 2. t For couronne-TABLE. Y

Sp'ecilcation ot Letters Iatent.

l Application med latch 15,1917. Seria1No.155,`017.

which form a part'of t The invention consists further in 'various details of construction .of the device,fcom".l vbination and arrangements of .parts which l will be hereinafter yfully described, shownin'" n,ex ressed as the sum of t e'powers of 2.V This 1s the fundamental purpose in the theor'yf, of numbers and is well known to one UNiTEDsTATEs PATENT OFFICE CLARENCE H. WALLAGE, or s'r. HELENA, cALIFonNIA. v

Patented Mal. 25, 1919.

example: If it is power of 2 plus the first power of 2.' Similarly 31 is equal to 16-l-8-1-4-1-2-1-2 which expressed in powers of 2 is 5|4|3|2+1+0.

:These two numbers can accordingly, be expressed as the sum ofthe finite Iseries of powers `of 2, as above clearly pointed out. We may then entirely disregard'the'modulus 2, 6a-=6 11, anda1=2+5+4|3+ l-i-O. These may be multiplied in the manner analogous to thel use of logarithme:

It is to be understood that each of these numbers of the product may be an exponent of 2. Accordingly the product 4is the 11th power of 2{21{ etc. Referring to Fig. 5

it is found that this product is the sum of 1024,'512, 256, 128, 64, s2, 16, s, 4,. 2, 1,6116 sum of which is 2046, which, as will be noted is the actual product of 66 3Lv desired to multiply two such numbers as 66 and 31, it is to be noted thatA 66`is equal to 64 plus 2 or the sixthv 'Referring to-Fig..1 insert: Specifically the invention consists of .a computing comprising aboard perforated wlth holes device `for the reception of pegs and in the provision of a longitudinally movable slide mounted in a slot in the surface of the board and containing .holes for the reception of transverse and diagonal lines with pegre- If'now we set the numbers 4, 3, 2,1, and

0 bymeans ofpegs in the right handvside of the board in Flg. 1 in column R at the pointsl z' designatedby .the numerals 4, 3, 2, 1, O, as clearly illustrated, likewise place on the lower row the pegs M and M in holes 1 to v 6 respectively. In multiplying as above noted vthe exponents` are added. This may f'bedone graphically by the following fprocess: Beginning with peg M9 following along Y f thehorizontalline on which said peg Ioccurs until peg is reached, since zero will not augment this`sum the peg 1 remainsl in its former position, Continuing on until the peg 6 is reached thissame peg will remain in the same posltion for again the-sum is not augmented. Then take exponent one,'follow `along the horizontal line in which this eg occurs until the intersection of the left diagunal-with peg M lis reached, place the peg P2 in the hole therein. Continue along the same line until the corresponding diagonal for pegM6 is reached and placca peg P7l in the hole. 'Following down along thevertical line of peg P2 it is found that this is'di- 4rectly above exponent 2 onthe lower line with respect to M and M, Similarly for pegs vP5 and P1". It is now to be'noted that pegs occur in this board at positions above numerals 10, 9, 8, 7, 6, 5, 4, 3, .2, 1, 0. It is to benoted that the numbers on the lower line of Fig. 1 correspond in position with, re-

spect to the powers exhibited on the lower line thereof to the powers of 2 given in the products above noted. Thus we are able to accomplish multiplication with the device herein described. A

Suppose it is desired to multiply two numbers such as 1023, which'may be readily .larger circles is as indicated in said F ig.l 2.

It 1s to be noted that some of the lines as measured from the longitudinal line contain more than one peg in cont-rast to the first example. In order to avoid multipli- We proceed as before cation in any line where two pegs occur, r

such as fork example in line 9, both these pegs may be removed and one placed in line 10, for it is to be noted that the number represented by line 10 is twice the number represented by line 9. In this way, byrmoving two pegs some one number will substitute one number in the next higher and we are able to multiply without partial products. If we continue in this way to eliminate the pegs according to the followingl schedule: Two pegs are found in line 3, remove them and place one in line 4:. None are left:

' and none higher. f 'j remaining pegs as found'fromy the "above v schedule `may-be placed in the upper rbwin Fig. 2 as is indicated bythe circles on said row. Thus it is seen that 'anyfproblemioff* multiplication may be readily accomplished' by the herein described method. Suppose one of the numbers in the above multiplica` reciprocal. Thus if 1'4 is the reciprocal of some number n the product of the reciprocal,

with-any number m the product it lis to'be remembered the 1' is ,-n 'and there- `sultant product is really 7%' `or the quotit.

the aid of a table of reciprocals. Another method of dividing is as follows:

less than a powerv of 2, the reciprocal of such a number may be expanded into a definite L, power series of which the first termi'sthe reciprocal of the next higher power of.,2, Athe second the zero thereof, the third term the cube rthereof and so forth. Take forexythus it is seen that division may begaccomplished by the device herein described with Suppose you have'anydivisor which isone lample the'number glfwhich is lnoted to be 32-1. 31T may accordingly be written 11- -l- (3%)2 -l- (315) (315V. and so forth. Such a number may be represented onthe l board as follows: 4For 31g, is the negative fifth 'power of 2, or in other words, the ifth power of Placing the peg in the hole v by the hole T5 and so forth. Again we have a reciprocal resolved in powers of 2, and this reciprocal may be multiplied with any desired number whatever with the same process as above outlined for the multiplication of integral numbers. It is, however, to

be noted that these powers are negative instead of positive and in the addition of these powers to those ofany multiplication it is necessary to proceed along the left hand diagonal instead of the right as is clear to one skilled in the art. Itfis to be noted that only afew terms of this reciprocal have been considered for time is too limited to consider more. A few terms are sufficient tov give re# sults for all practical purposes. It is of course understood that if l desired, more terms may be taken anda correspondingly more accurate result obtained.

We. can also divide directly when thedivisor is an integral power of 2 for setting up such a number on the negative scale as above we have only one exponent to deal with vand multiplylng (which is no more than dividing when the 'negative quotient is considered) we proceed as before and thereby obtain a quotient.

vWhat I .claim to be new is 1. As a means tofacilitate calculating, a

' device Awhich graphically'or mechanically gives the numbers in the form of the integral exponents of a geometrical 'series of powers of 2, said series expressing any desired numbers the sum of which powers will give a series which is the product of the numbers so expressed'. f

2. As a means to facilitate calculating, a device which graphically or mechanically gives the numbers in the form of the integral exponents of a geometrical series of powers of 2, said series being capable of expressing any desired number the difference of which powers will give the quotient of the numbers expressed as a series. l

3. In a device of the class described, means for graphically or mechanically representing any number as a geometrical series of the powers of 2 by means of locations predetermined by the series.

4. A computing device comprising a board divided by transverse, horizontal and diagonal linesl with peg holes at intersections' of the line, a transversely movable slide mounted in a slot in the board and having peg holes identified by numerals increasing in regular order in opposite directions, the horizontal lines of peg holes being 4designated lfrom left to right by numerals from 1 upward, the peg holes at the bottom being -deslgnated by numerals in reverse order, the horizontal lines being designated at the left by numerals from 1 to 11 consecutively and the horizontallines at the right of the board being identified by reversely arranged numerals from 11 to zero. In testimony whereof I hereunto aflix my signature in presence of two witnesses.

CLARENCE H. WALLACE. 

